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Differential Geometry Intro

Differential Geometry Intro. Catalog Description. MAT 4140 – 3 hours [On-Demand] Differential Geometry

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Differential Geometry Intro

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  1. Differential Geometry Intro

  2. Catalog Description • MAT 4140 – 3 hours [On-Demand] • Differential Geometry • Bulletin Description: This is an introductory course in the differential geometry of curves and surfaces in space, presenting both theoretical and computational components, intrinsic and extrinsic viewpoints, and numerous applications. The geometry of space-time will also be considered. Prerequisite: MAT 2130.

  3. Background • Geometry: Riemannian Geometry PhD • Physics: Special relativity with Ralph Alpher, one of the creators of the big bang. • Research: Geometry of Orbifolds, Mathematics and Society: Popular Culture, Women & Minorities…

  4. Course Goals • To develop geometric problem solving skills and 3-D spatial visualization skills. • To develop a greater appreciation for connections between various disciplines of mathematics, including geometry, linear algebra, complex analysis, and differential equations, along with an introduction to these subjects as they apply to differential geometry. • To understand the importance of differential geometry in various scientific fields, including physics.

  5. Topics • Geometry of curves in space, including Frenet formulas • Theory of surfaces, including curvature, geodesics, and metrics • Geometry of space-time and applications to general relativity (as time allows)

  6. Culmination of Early Ideas • At a very early age, children develop a very rich `visual intelligence' in terms of perception and experiences. They have questions and lots of these questions and explorations can be connected to geometry if we use the right types of physical and visual presentations. They have developed advanced skills for which the precise vocabulary is 3-D differential geometry and differential topology [28 (Hoffmann, 1998), 36 (Koenderink, 1990)]...we should connect with these abilities. (Whiteley, 1999)

  7. Rich History and New Relevance • Machinery Design • Classification of Spaces • Fundamental Forces in the Universe • Study of DNA • Medical Imaging • Computer Graphics

  8. Approaches • Intuitive • Calculable • Useful • Interdisciplinary: unification of many topics including geometry, spatial visualization, calculus, linear algebra, differential equations, and complex variables along with various topics from the sciences, including physics

  9. Smokestack Problem • Frank Morgan received a call from a company constructing a huge smokestack, which required the attachment of a spiraling strip or stake for structural support.

  10. Strake cut out of flat metal pieces • What inner radius would make it fit best?

  11. Soap Bubbles • Pressure versus surface tension • What shape does it take on, and why?

  12. Inconsistency in Mercury’s Orbit • Newton used solar gravitational attraction and calculus to explain Kepler’s elliptical planetary orbits. • The orbit rotated (precessed) at an unexpected rate • General relativity and the geometry of space-time

  13. Curves • Some of the results were motivated by the desire for more accurate clocks: I have been occupied with a new discovery... In order to make my clock even more exact.... What, however, I never had expected I would discover, I have now hit upon, the undoubtedly true shape of curves... I determined it by geometric reasoning. (C. Huygens December 1659)

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